DFA
Amy Atwater, Emma Curtis, and Ingrid Lundeen
December 8, 2016
DISCRIMINANT FUNCTION ANALYSIS
INTRODUCTION
MATHY SECTION
“Bootstraps bootstraps” - The scene in Pirates of the Caribbean, the movie about statistics on the high seas, where William Turner learns about a common resampling method - bootstrapping.
“No, not that knife, Will!” - The scene in Pirates of the Caribbean where Professor Jack Sparrow, the swashbuckling statistician, teaches young William Turner about another common resampling method - jackknifing.
EXAMPLES
Davis and McHorse, 2013
Palaeontologia Electronica
A method for improved identification of postcrania from mammalian fossil assemblages: multivariate discriminant function analysis of camelid astragali.
Aepycamelus, one of these North American camelids
library(curl) #needed to read in csv file from github
camels <- read.csv(curl("https://raw.githubusercontent.com/ilundeen/nothing-to-see-here/master/camel.csv"), header=TRUE)
#Read in the .csv file
head(camels)## Mus Loc
## 1 AMNH ""Alachula Clays"" Mixon's Bone Bed, Levy Co., FL
## 2 AMNH ""Alachula Clays"" Mixon's Bone Bed, Levy Co., FL
## 3 AMNH ""Alachula Clays"" Mixon's Bone Bed, Levy Co., FL
## 4 AMNH ""Alachula Clays"" Mixon's Bone Bed, Levy Co., FL
## 5 AMNH ""Alachula Clays"" Mixon's Bone Bed, Levy Co., FL
## 6 AMNH ""Alachula Clays"" Mixon's Bone Bed, Levy Co., FL
## Sp._ Genus Genus..2. Taxon Side
## 1 FLOR-121-2193 (1941) Aepycamelus Aepycamelus Aepycamelus major R
## 2 FLOR 28 #489 (1940) Aepycamelus Aepycamelus Aepycamelus major R
## 3 FLOR-152-2490 (1942) Aepycamelus Aepycamelus Aepycamelus major R
## 4 FLOR-146-2450 (1942) Aepycamelus Aepycamelus Aepycamelus major R
## 5 FLOR 30 #535 (1940) Aepycamelus Aepycamelus Aepycamelus major R
## 6 FLOR 2 #31 (1940) Aepycamelus Aepycamelus Aepycamelus major R
## LM TD TI TP LL WD WI LI
## 1 95.13 39.90 54.34 45.32 105.67 69.38 74.50 80.24
## 2 95.25 38.22 54.65 45.25 105.20 69.73 75.04 80.36
## 3 100.35 39.63 54.86 44.27 109.10 69.83 72.23 83.84
## 4 98.09 39.90 53.26 42.44 105.57 73.83 76.32 82.74
## 5 94.10 40.75 51.95 42.16 101.15 67.47 68.25 80.41
## 6 90.31 34.36 47.68 40.34 100.38 67.74 70.37 78.14
## Notes Prin.Comp.1 Prin.Comp.2
## 1 4.726898 0.08269429
## 2 Two right and two left with Same Field No 4.664651 -0.16147633
## 3 4.882082 -0.05180152
## 4 4.828685 0.02919387
## 5 Two with Same Field No 4.232418 0.36355821
## 6 Three with Same Field No 3.570367 -0.45557335
## Prin.Comp.3
## 1 0.11249259
## 2 0.06565884
## 3 0.17052671
## 4 -0.32338462
## 5 0.14464094
## 6 -0.23328700
Abbreviations: LM= medial length; LI= intermediate length; LL= lateral length; TD= distal thickness; TI= intermediate thickness; TP= proximal thickness; WD= distal width; WI= intermediate width
Linear Discriminant Function
Here we’re going to use the lda() function contained in the {MASS} package
Note: There are other packages that run linear discriminant analyses including:
{DiscriMiner}
{mda}
{dawai}
{rrlda}
{sparsediscrim}
library(MASS) #lda() and qda() can both be found in this package
camel.lda <- lda(Genus..2. ~ LM + TD + TI + TP + LL + WD + WI + LI, data=camels, prior= c(1,1,1,1,1)/5, na.action="na.omit") # Note that we're using + instead of * becuase we aren't modeling any variable interactions here
# Prior is specified to be uninformative; if not included, default prior is set by relative abundances in the training set.
camel.lda## Call:
## lda(Genus..2. ~ LM + TD + TI + TP + LL + WD + WI + LI, data = camels,
## prior = c(1, 1, 1, 1, 1)/5, na.action = "na.omit")
##
## Prior probabilities of groups:
## Aepycamelus Alforjas Hemiauchenia Megatylopus Procamelus
## 0.2 0.2 0.2 0.2 0.2
##
## Group means:
## LM TD TI TP LL WD
## Aepycamelus 92.03708 37.36500 49.18667 41.68917 98.84625 66.96438
## Alforjas 58.12042 27.79625 33.21125 28.46583 63.85958 42.73292
## Hemiauchenia 50.45350 23.21675 27.96225 24.00013 54.34913 35.70287
## Megatylopus 79.62783 33.81348 43.73391 37.09565 87.04217 60.14348
## Procamelus 66.68000 28.91333 35.94000 33.16167 73.38333 47.09333
## WI LI
## Aepycamelus 68.94229 78.17250
## Alforjas 44.22375 49.25000
## Hemiauchenia 38.46363 41.53437
## Megatylopus 61.44304 68.53522
## Procamelus 49.16167 55.34667
##
## Coefficients of linear discriminants:
## LD1 LD2 LD3 LD4
## LM 0.17843479 -0.26154962 0.612728795 -0.22214235
## TD -0.08833182 0.06511872 -0.237187959 -0.37465125
## TI 0.03972081 0.28547071 0.005929533 -0.28948794
## TP -0.10287848 -0.52113821 -0.301629648 -0.04092927
## LL -0.03361356 -0.29001249 -0.180872844 0.30903264
## WD 0.08460694 0.20045633 -0.236866116 0.32094209
## WI -0.04463453 0.04310350 0.243654169 0.09190961
## LI 0.23894225 0.48890370 -0.251019220 -0.15398852
##
## Proportion of trace:
## LD1 LD2 LD3 LD4
## 0.9431 0.0377 0.0149 0.0044Displayed in this output
Call:
The formula that was fitted
Prior probabilities of groups:
The probability of randomly selecting a sample falling within each group within our dataset. If unspecified in our call, this defaults to the actual proportion of the total of each user-defined group within our data. If specified, these are whatever we indicated as our priors.
Group means:
Table of average values for each of our variables by group. Useful for determining if any of our groups seem distinctive or somewhat off.
Coefficients of linear discriminants:
Coefficients of the discriminant function. There will be # of groups - 1 linear discriminants. Here we have 5 groups so we’ll have 4 linear discriminants.
Proportion of trace:
The proportion of trace shows the proportion of variance between our groups explained by the linear discriminants.
Now we are interested in how well our model reassigns our samples to our groups.
The predict() function allows us to check how successfully the linear function we just derived classifies our data into groups.
camel.p <- predict(camel.lda)
camel.p## $class
## [1] Aepycamelus Aepycamelus Aepycamelus Aepycamelus Aepycamelus
## [6] Aepycamelus Aepycamelus Aepycamelus Aepycamelus Aepycamelus
## [11] Aepycamelus Aepycamelus Aepycamelus Megatylopus Aepycamelus
## [16] Aepycamelus Aepycamelus Megatylopus Aepycamelus Aepycamelus
## [21] Aepycamelus Aepycamelus Aepycamelus Aepycamelus Aepycamelus
## [26] Aepycamelus Aepycamelus Aepycamelus Aepycamelus Megatylopus
## [31] Aepycamelus Aepycamelus Aepycamelus Aepycamelus Aepycamelus
## [36] Aepycamelus Aepycamelus Aepycamelus Aepycamelus Aepycamelus
## [41] Aepycamelus Aepycamelus Aepycamelus Aepycamelus Aepycamelus
## [46] Megatylopus Aepycamelus Aepycamelus Hemiauchenia Hemiauchenia
## [51] Hemiauchenia Hemiauchenia Hemiauchenia Hemiauchenia Hemiauchenia
## [56] Hemiauchenia Hemiauchenia Hemiauchenia Hemiauchenia Hemiauchenia
## [61] Hemiauchenia Hemiauchenia Hemiauchenia Hemiauchenia Hemiauchenia
## [66] Hemiauchenia Hemiauchenia Hemiauchenia Hemiauchenia Hemiauchenia
## [71] Hemiauchenia Hemiauchenia Hemiauchenia Hemiauchenia Hemiauchenia
## [76] Hemiauchenia Hemiauchenia Hemiauchenia Hemiauchenia Hemiauchenia
## [81] Hemiauchenia Hemiauchenia Hemiauchenia Hemiauchenia Hemiauchenia
## [86] Hemiauchenia Hemiauchenia Hemiauchenia Hemiauchenia Hemiauchenia
## [91] Hemiauchenia Hemiauchenia Hemiauchenia Hemiauchenia Hemiauchenia
## [96] Hemiauchenia Hemiauchenia Hemiauchenia Hemiauchenia Hemiauchenia
## [101] Hemiauchenia Hemiauchenia Hemiauchenia Hemiauchenia Hemiauchenia
## [106] Hemiauchenia Hemiauchenia Hemiauchenia Hemiauchenia Hemiauchenia
## [111] Hemiauchenia Hemiauchenia Hemiauchenia Hemiauchenia Hemiauchenia
## [116] Hemiauchenia Hemiauchenia Hemiauchenia Hemiauchenia Hemiauchenia
## [121] Hemiauchenia Hemiauchenia Hemiauchenia Megatylopus Aepycamelus
## [126] Megatylopus Megatylopus Megatylopus Megatylopus Megatylopus
## [131] Megatylopus Megatylopus Megatylopus Megatylopus Megatylopus
## [136] Megatylopus Megatylopus Megatylopus Megatylopus Megatylopus
## [141] Megatylopus Megatylopus Megatylopus Megatylopus Megatylopus
## [146] Megatylopus Alforjas Alforjas Alforjas Alforjas
## [151] Alforjas Alforjas Alforjas Alforjas Alforjas
## [156] Alforjas Alforjas Alforjas Alforjas Alforjas
## [161] Alforjas Alforjas Alforjas Alforjas Alforjas
## [166] Hemiauchenia Alforjas Alforjas Alforjas Alforjas
## [171] Procamelus Procamelus Procamelus Procamelus Procamelus
## [176] Procamelus Hemiauchenia Hemiauchenia Hemiauchenia Hemiauchenia
## [181] Hemiauchenia
## Levels: Aepycamelus Alforjas Hemiauchenia Megatylopus Procamelus
##
## $posterior
## Aepycamelus Alforjas Hemiauchenia Megatylopus Procamelus
## 1 9.999704e-01 4.844915e-30 1.551687e-44 2.961259e-05 9.607793e-19
## 2 9.999878e-01 1.717864e-31 1.021650e-45 1.215902e-05 5.361055e-20
## 3 1.000000e+00 2.538645e-40 3.216243e-56 2.257359e-09 5.636456e-27
## 4 9.999998e-01 2.698264e-38 1.491667e-53 2.349458e-07 1.194647e-27
## 5 9.999840e-01 1.297946e-30 3.752229e-46 1.597350e-05 9.370914e-22
## 6 9.968717e-01 3.407794e-28 4.525622e-41 3.128339e-03 3.615614e-18
## 7 9.993273e-01 3.407916e-23 3.502462e-36 6.727203e-04 1.597663e-13
## 8 9.999995e-01 5.660789e-37 7.123536e-53 4.564793e-07 2.967987e-26
## 9 9.999395e-01 4.621318e-31 1.062022e-46 6.053038e-05 4.068939e-23
## 10 9.718730e-01 8.558465e-21 3.023656e-35 2.812697e-02 5.628379e-14
## 11 9.907861e-01 3.341846e-26 5.817952e-40 9.213877e-03 5.179408e-17
## 12 8.466563e-01 2.200975e-23 2.241242e-34 1.533437e-01 5.821150e-15
## 13 9.214822e-01 3.040378e-22 4.290575e-35 7.851783e-02 1.646814e-12
## 14 3.314466e-02 3.806282e-16 2.838102e-26 9.668553e-01 3.307543e-08
## 15 1.000000e+00 2.998566e-38 1.243793e-52 1.444728e-08 4.211514e-27
## 16 9.999979e-01 4.128674e-32 1.598954e-45 2.070713e-06 2.698287e-19
## 17 9.841773e-01 8.488111e-24 7.440040e-38 1.582267e-02 2.732087e-17
## 18 4.186575e-01 1.217537e-18 3.788419e-31 5.813425e-01 2.718426e-12
## 19 9.998583e-01 2.672689e-27 1.564714e-41 1.417059e-04 8.744445e-18
## 20 9.999839e-01 6.755463e-32 3.118857e-47 1.607494e-05 2.831473e-25
## 21 9.990397e-01 1.493142e-28 3.502684e-43 9.603445e-04 1.276271e-20
## 22 9.995367e-01 5.641398e-28 5.184960e-40 4.633498e-04 2.636280e-18
## 23 9.999826e-01 1.285321e-27 1.784796e-41 1.737729e-05 1.740111e-18
## 24 9.888925e-01 1.267646e-21 1.686873e-33 1.110746e-02 5.214603e-13
## 25 9.795862e-01 3.378327e-25 4.183345e-39 2.041380e-02 5.944626e-17
## 26 9.999955e-01 6.326949e-32 2.086717e-45 4.536273e-06 3.714380e-23
## 27 9.997259e-01 2.545608e-27 7.286249e-40 2.740878e-04 1.978811e-17
## 28 9.980127e-01 3.918939e-26 2.428176e-40 1.987276e-03 5.555670e-15
## 29 9.999984e-01 1.031963e-33 7.263821e-49 1.614898e-06 1.817173e-21
## 30 2.964244e-03 2.481835e-12 2.379289e-21 9.970357e-01 4.003052e-08
## 31 9.997722e-01 8.886177e-29 1.420803e-42 2.277788e-04 3.062552e-18
## 32 9.999932e-01 1.194180e-28 1.090744e-41 6.766740e-06 1.116083e-15
## 33 9.999954e-01 5.747254e-30 1.480114e-42 4.633990e-06 2.205885e-18
## 34 9.999951e-01 1.314488e-34 3.318921e-49 4.882879e-06 2.259321e-23
## 35 9.999933e-01 4.119459e-34 7.918727e-50 6.730331e-06 1.963192e-23
## 36 9.944174e-01 6.549469e-29 5.402559e-44 5.582584e-03 1.205835e-19
## 37 9.999995e-01 5.713730e-39 5.872208e-56 4.600350e-07 5.557007e-27
## 38 9.995681e-01 6.234291e-27 3.705627e-40 4.318577e-04 3.459639e-17
## 39 9.999509e-01 8.315521e-28 1.466432e-39 4.908144e-05 4.241122e-18
## 40 9.998209e-01 4.768070e-28 1.033489e-40 1.791148e-04 1.390129e-18
## 41 9.611418e-01 3.018534e-24 1.186891e-37 3.885816e-02 4.240409e-16
## 42 9.999959e-01 1.513868e-34 1.970481e-48 4.056217e-06 8.840115e-23
## 43 9.999097e-01 3.845152e-30 1.360622e-45 9.033376e-05 1.836124e-20
## 44 9.999812e-01 1.653777e-30 4.484338e-45 1.878450e-05 4.446176e-18
## 45 9.998382e-01 1.035170e-26 1.269699e-38 1.618110e-04 3.996706e-17
## 46 5.643835e-02 1.444382e-13 8.988535e-21 9.435616e-01 1.238173e-08
## 47 9.979631e-01 1.823145e-23 3.759512e-36 2.036872e-03 2.633911e-12
## 48 1.000000e+00 8.700716e-39 4.463512e-52 9.304585e-09 7.517745e-28
## 49 3.158663e-41 1.154398e-02 9.884559e-01 2.860928e-22 9.744667e-08
## 50 7.613699e-43 6.215998e-03 9.937839e-01 8.463408e-23 9.387805e-08
## 51 9.014835e-42 1.285433e-02 9.871456e-01 2.207806e-22 3.711289e-08
## 52 2.300245e-46 2.700665e-03 9.972993e-01 7.256526e-26 1.103074e-08
## 53 1.226134e-36 3.290832e-01 6.707435e-01 3.072346e-19 1.732953e-04
## 54 1.734107e-42 9.665540e-03 9.903343e-01 2.418185e-23 1.361696e-07
## 55 3.119836e-45 2.541181e-03 9.974588e-01 7.885918e-25 3.797379e-08
## 56 1.965234e-37 1.090908e-02 9.890505e-01 2.907317e-20 4.038711e-05
## 57 1.280745e-46 1.399654e-03 9.986003e-01 6.924089e-26 9.133898e-09
## 58 1.292874e-45 4.974239e-04 9.995026e-01 8.188209e-26 1.116458e-08
## 59 2.055651e-48 7.056397e-04 9.992944e-01 3.750725e-27 4.761709e-09
## 60 9.700363e-49 7.178530e-04 9.992821e-01 2.057971e-27 1.553498e-10
## 61 9.332908e-45 9.063085e-05 9.999093e-01 8.462997e-26 3.864146e-08
## 62 3.382356e-44 3.930179e-04 9.996069e-01 1.130156e-24 1.203394e-07
## 63 6.853719e-39 1.346714e-02 9.865319e-01 2.222356e-21 9.626364e-07
## 64 4.019802e-36 2.418813e-02 9.723969e-01 4.297559e-19 3.414996e-03
## 65 2.909333e-43 9.849687e-03 9.901500e-01 7.570491e-24 2.980833e-07
## 66 4.909913e-40 1.260561e-02 9.873940e-01 1.558741e-21 4.371844e-07
## 67 3.328044e-38 9.035879e-02 9.096353e-01 7.198566e-20 5.911532e-06
## 68 4.990087e-40 1.594826e-02 9.840504e-01 1.148170e-21 1.376069e-06
## 69 1.648481e-47 7.549776e-04 9.992450e-01 1.285764e-26 4.341199e-09
## 70 3.185208e-40 7.216042e-03 9.927827e-01 5.216654e-22 1.231133e-06
## 71 2.614839e-41 3.489572e-02 9.651033e-01 2.545465e-22 9.473745e-07
## 72 7.763610e-42 1.123984e-02 9.887599e-01 1.400511e-22 2.684084e-07
## 73 1.922832e-44 3.745614e-03 9.962543e-01 2.512281e-24 6.151467e-08
## 74 1.552502e-39 1.382754e-02 9.861599e-01 1.295530e-21 1.253946e-05
## 75 4.177382e-40 3.839396e-02 9.616029e-01 8.742789e-22 3.125964e-06
## 76 8.000697e-42 2.038261e-02 9.796171e-01 3.036938e-22 3.032869e-07
## 77 9.134725e-49 4.451668e-04 9.995548e-01 1.278139e-27 1.592993e-08
## 78 2.194899e-42 2.839613e-03 9.971595e-01 1.650161e-23 8.951744e-07
## 79 1.534604e-43 2.157400e-03 9.978422e-01 4.059954e-24 3.922163e-07
## 80 6.169540e-44 4.505124e-03 9.954948e-01 4.202270e-24 7.618489e-08
## 81 4.292244e-42 9.659713e-03 9.903402e-01 4.168338e-23 5.072180e-08
## 82 1.786377e-35 7.142673e-02 9.285124e-01 3.820806e-18 6.086283e-05
## 83 3.595879e-35 2.594912e-02 9.740019e-01 2.004186e-18 4.894340e-05
## 84 6.167949e-39 8.641752e-04 9.991351e-01 1.761492e-21 7.041617e-07
## 85 6.370821e-48 9.496445e-04 9.990504e-01 7.772693e-27 1.981942e-09
## 86 9.448946e-45 2.968526e-03 9.970314e-01 8.597890e-25 4.213738e-08
## 87 2.288181e-41 8.930957e-03 9.910689e-01 2.682768e-22 1.059415e-07
## 88 3.016217e-35 2.076962e-01 7.919794e-01 3.986638e-18 3.244463e-04
## 89 3.308794e-41 1.952415e-02 9.804759e-01 3.197554e-22 8.878747e-10
## 90 1.660605e-55 2.938622e-05 9.999706e-01 3.973187e-32 3.350098e-12
## 91 3.351038e-40 3.971185e-02 9.602880e-01 8.600887e-21 1.299389e-07
## 92 7.809404e-47 1.388971e-03 9.986110e-01 5.602638e-26 2.144930e-09
## 93 1.067347e-41 4.082692e-02 9.591730e-01 5.128479e-22 6.472396e-08
## 94 3.495565e-40 6.326476e-03 9.936732e-01 1.486173e-21 2.895273e-07
## 95 2.075851e-40 6.653172e-03 9.933460e-01 1.075570e-21 7.905739e-07
## 96 7.097837e-43 1.296138e-02 9.870386e-01 3.979685e-23 6.505865e-08
## 97 4.380311e-46 5.375687e-04 9.994624e-01 3.472361e-26 1.207663e-08
## 98 3.917714e-46 5.672314e-03 9.943277e-01 7.484719e-26 8.510860e-09
## 99 6.081755e-47 2.872532e-04 9.997127e-01 5.993873e-27 8.175713e-10
## 100 1.629139e-41 1.269903e-02 9.873010e-01 2.955772e-22 1.282685e-08
## 101 3.313216e-40 6.224058e-03 9.937756e-01 1.697332e-21 3.666385e-07
## 102 7.183497e-44 6.599263e-04 9.993401e-01 7.323132e-25 7.626642e-09
## 103 1.027827e-44 8.354272e-04 9.991646e-01 5.774903e-25 7.978117e-09
## 104 5.980932e-42 4.706396e-02 9.529360e-01 3.748933e-22 3.592334e-08
## 105 5.233834e-42 1.025621e-01 8.974379e-01 6.365676e-22 9.872398e-09
## 106 1.252153e-38 2.547841e-02 9.745209e-01 1.622665e-20 6.560682e-07
## 107 8.456719e-42 1.818939e-03 9.981809e-01 4.457180e-23 1.457624e-07
## 108 4.464091e-41 1.309280e-02 9.869067e-01 3.672847e-22 4.679410e-07
## 109 6.613295e-42 4.078998e-03 9.959207e-01 3.415471e-23 2.688247e-07
## 110 7.386150e-40 1.845814e-02 9.815414e-01 2.777932e-21 4.758537e-07
## 111 7.743576e-36 8.973569e-02 9.101683e-01 5.537188e-18 9.601490e-05
## 112 3.179400e-48 9.915161e-04 9.990085e-01 6.793027e-27 3.036605e-10
## 113 1.323714e-40 1.036107e-02 9.896368e-01 1.325311e-22 2.083214e-06
## 114 1.853313e-46 3.601099e-03 9.963989e-01 8.813901e-26 1.134745e-08
## 115 2.360744e-35 1.062618e-01 8.937176e-01 2.017879e-17 2.061084e-05
## 116 1.984319e-31 4.953528e-01 5.044240e-01 8.993629e-15 2.231931e-04
## 117 2.689026e-41 3.543619e-03 9.964564e-01 8.544390e-22 2.169917e-09
## 118 3.294378e-41 1.519489e-03 9.984805e-01 1.531278e-22 2.667304e-08
## 119 9.658341e-36 1.238155e-01 8.761784e-01 1.296609e-17 6.154220e-06
## 120 2.215479e-42 1.160872e-02 9.883908e-01 2.337251e-22 4.320740e-07
## 121 1.402317e-31 3.517850e-01 6.480698e-01 8.494206e-15 1.451924e-04
## 122 5.115513e-33 2.555010e-01 7.444845e-01 2.105657e-16 1.455108e-05
## 123 6.025586e-38 7.499463e-03 9.924994e-01 1.942974e-20 1.118505e-06
## 124 5.388233e-03 7.124665e-17 2.066512e-28 9.946118e-01 5.002372e-10
## 125 5.064916e-01 2.342418e-17 2.984182e-28 4.935084e-01 1.016891e-09
## 126 2.373835e-05 5.709888e-12 2.924187e-21 9.999762e-01 4.834147e-08
## 127 2.001900e-07 1.115401e-06 1.120088e-14 9.998539e-01 1.447848e-04
## 128 2.535479e-02 2.544419e-12 4.253156e-21 9.741963e-01 4.489232e-04
## 129 2.564816e-06 5.085328e-07 2.289625e-14 9.999685e-01 2.841175e-05
## 130 1.224552e-08 4.657967e-06 1.439181e-13 9.991802e-01 8.151517e-04
## 131 1.468865e-03 4.468622e-15 7.264291e-26 9.985311e-01 1.896528e-11
## 132 3.007936e-06 1.462358e-10 3.236933e-20 9.999963e-01 6.702526e-07
## 133 2.286600e-05 6.126402e-13 2.417466e-24 9.999771e-01 5.466475e-08
## 134 2.469074e-03 6.018013e-15 5.411001e-26 9.975309e-01 1.654127e-09
## 135 4.633080e-01 2.125144e-19 3.960439e-33 5.366920e-01 6.832852e-13
## 136 3.629386e-05 3.583129e-11 4.720791e-21 9.999444e-01 1.929279e-05
## 137 8.282796e-06 2.202815e-10 4.000926e-19 9.999912e-01 5.560342e-07
## 138 4.631509e-03 1.325165e-16 2.233547e-29 9.953685e-01 6.206525e-12
## 139 6.158568e-02 4.081173e-19 5.410255e-31 9.384143e-01 6.490373e-12
## 140 1.217931e-05 2.885628e-13 4.264610e-24 9.999878e-01 1.721591e-08
## 141 2.343171e-03 2.020134e-13 3.017892e-23 9.976564e-01 4.078593e-07
## 142 1.138823e-06 7.127442e-09 2.735245e-18 9.999937e-01 5.135563e-06
## 143 3.697504e-06 6.292640e-10 5.859615e-18 9.999957e-01 5.745918e-07
## 144 3.621620e-06 2.660634e-09 3.659842e-16 9.999778e-01 1.858063e-05
## 145 2.546308e-01 8.546845e-17 1.148367e-27 7.453692e-01 3.337056e-10
## 146 3.817344e-03 1.575492e-13 8.380221e-23 9.961825e-01 1.408148e-07
## 147 3.392227e-28 9.585887e-01 1.147467e-02 3.936894e-13 2.993664e-02
## 148 7.720373e-27 9.950458e-01 3.160377e-03 6.542372e-12 1.793798e-03
## 149 3.866181e-26 9.784145e-01 1.346470e-02 1.808914e-11 8.120756e-03
## 150 9.019656e-31 8.997853e-01 9.767832e-02 1.344859e-13 2.536346e-03
## 151 8.918218e-33 8.255261e-01 1.744719e-01 2.727409e-14 2.079869e-06
## 152 4.969479e-32 9.665775e-01 3.321080e-02 7.529640e-15 2.116892e-04
## 153 4.483072e-28 9.898630e-01 1.012559e-02 7.089732e-12 1.145585e-05
## 154 2.925602e-25 9.980096e-01 6.406513e-04 4.110445e-11 1.349740e-03
## 155 5.372361e-28 9.654799e-01 2.434858e-02 7.570110e-13 1.017149e-02
## 156 3.078141e-24 9.503800e-01 4.650927e-04 7.081112e-10 4.915489e-02
## 157 3.712390e-32 9.850819e-01 1.077938e-02 1.331494e-15 4.138678e-03
## 158 5.917755e-26 9.950974e-01 3.467912e-04 2.405039e-11 4.555801e-03
## 159 2.621449e-25 9.384130e-01 2.029116e-03 1.072417e-11 5.955788e-02
## 160 6.262371e-26 9.998301e-01 6.285331e-05 4.176824e-11 1.070031e-04
## 161 2.719705e-28 9.972538e-01 5.991341e-04 2.251328e-13 2.147108e-03
## 162 2.608487e-28 9.678712e-01 2.903900e-02 5.954483e-13 3.089832e-03
## 163 1.293099e-28 9.648031e-01 3.492675e-02 6.719037e-12 2.701021e-04
## 164 1.533602e-29 9.060450e-01 7.507726e-02 2.653080e-14 1.887770e-02
## 165 1.041524e-33 9.897859e-01 9.949296e-03 7.531293e-17 2.647839e-04
## 166 9.772394e-35 2.285891e-01 7.707833e-01 6.860435e-18 6.275310e-04
## 167 2.683994e-34 8.623258e-01 1.376260e-01 4.531440e-17 4.814337e-05
## 168 4.792984e-24 8.772098e-01 1.270115e-02 1.765347e-09 1.100891e-01
## 169 5.443102e-25 9.927115e-01 6.751208e-05 5.529889e-10 7.221032e-03
## 170 4.835442e-21 9.835318e-01 1.432224e-03 3.664513e-08 1.503592e-02
## 171 4.670639e-12 3.141584e-04 3.703818e-08 3.005677e-04 9.993852e-01
## 172 7.636454e-22 2.061364e-02 9.545914e-08 4.044623e-09 9.793863e-01
## 173 1.148094e-21 1.834947e-03 2.630538e-06 3.081541e-10 9.981624e-01
## 174 2.652394e-19 5.216881e-03 5.612678e-07 4.859240e-08 9.947825e-01
## 175 7.091411e-19 5.469252e-03 1.177136e-07 3.207263e-08 9.945306e-01
## 176 1.029773e-17 1.871275e-04 4.271650e-08 1.599252e-07 9.998127e-01
## 177 1.257061e-41 1.204115e-01 8.795871e-01 5.520025e-22 1.412759e-06
## 178 9.178163e-37 1.411299e-01 8.588699e-01 5.505490e-18 1.518752e-07
## 179 3.161769e-44 1.746215e-03 9.982538e-01 6.493048e-24 8.395166e-09
## 180 3.793476e-49 2.112674e-03 9.978873e-01 4.732750e-28 2.968742e-10
## 181 3.587879e-44 4.202734e-02 9.579727e-01 1.513333e-23 6.778392e-09
##
## $x
## LD1 LD2 LD3 LD4
## 1 7.88316567 -1.34812481 -0.164964364 -0.517977536
## 2 8.12247147 -1.07552470 0.433549135 -0.003859773
## 3 9.85141321 -1.27768507 1.233567419 -1.242550331
## 4 9.56070722 1.29199756 1.290937966 0.434597623
## 5 8.16437910 0.68284560 -0.354258972 -2.396851848
## 6 7.48188475 0.24652248 0.524455516 2.542795432
## 7 6.41697810 -1.57102297 0.370360798 -2.578869419
## 8 9.37803535 0.70466735 0.295903014 -0.275329852
## 9 8.33149107 1.85964741 -0.486511040 -1.413293315
## 10 6.26672407 0.36607595 -1.543698870 -3.242438469
## 11 7.22189570 0.23246544 -0.640230366 1.152497080
## 12 6.40850050 0.97246939 1.420831827 2.919448539
## 13 6.33914126 -1.14475707 -0.814352294 1.008768745
## 14 4.67295336 -0.74290281 0.288558381 1.428234049
## 15 9.35420475 0.55982498 2.754002027 -1.259064617
## 16 8.07187824 -2.13039005 1.416877812 0.003627276
## 17 6.83736566 1.74337044 -0.583747742 -1.520775540
## 18 5.64471541 0.94297293 -0.611189529 -0.776095522
## 19 7.37272801 -0.45957287 -0.066583874 -1.800737222
## 20 8.46300384 3.18496203 0.619974733 -2.405188431
## 21 7.77541357 1.63760454 -0.382131316 -0.229268064
## 22 7.29088469 0.34987775 1.955740967 1.061290235
## 23 7.34213606 -0.27093477 0.989518852 -3.346290920
## 24 6.07054679 -0.49132461 0.961668950 -1.114696182
## 25 7.07616175 0.79854287 -0.835274770 0.600838848
## 26 8.16398633 1.50147107 2.268542666 -1.387176960
## 27 7.19219528 -0.29087829 1.427626212 -0.051362390
## 28 7.16590786 -1.89017293 -1.282977842 0.430590630
## 29 8.61324093 -1.47003277 -0.009429692 -0.355044466
## 30 3.69709270 1.42789278 1.378767435 -1.023137041
## 31 7.61904031 -0.64026788 0.136609210 0.396451095
## 32 7.33997379 -3.26335417 1.105096804 -1.148875736
## 33 7.58477804 -1.51010472 2.197247987 -0.922970914
## 34 8.79317060 0.06765260 0.671272214 1.089290809
## 35 8.84257420 0.17838831 -0.470630951 0.097513048
## 36 7.92210910 0.78246578 -1.634300376 1.906761450
## 37 9.88848434 0.07370540 -0.989898867 1.110536232
## 38 7.20739104 -0.38311361 0.602509967 -0.403880484
## 39 7.15501922 -0.05983013 2.782998361 -0.814012889
## 40 7.35529106 0.25621323 1.603691941 -0.170637908
## 41 6.84064934 0.75588876 -0.549695322 0.737495599
## 42 8.67943309 -0.29532294 1.304708717 1.551019017
## 43 8.08776143 0.09337880 -0.975246319 -0.801203737
## 44 7.94733041 -2.27140754 -0.396893421 -0.090554976
## 45 6.98396859 -0.19236152 2.240505873 -0.712953286
## 46 3.90324823 1.61997024 4.076872813 -0.423030648
## 47 6.43509515 -2.53426774 0.018225172 -0.553807622
## 48 9.33167673 1.11868852 4.118945632 -0.715727651
## 49 -6.12642848 0.89240305 1.004835684 -0.232828681
## 50 -6.36978790 0.65359640 0.450175591 1.248146439
## 51 -6.20284658 1.24333470 0.776802414 -0.040997936
## 52 -7.02397477 0.24760688 0.399488374 -0.153959347
## 53 -5.43758817 -0.51102086 0.027969621 -1.207265395
## 54 -6.37427499 0.34839073 0.792321757 -0.553402924
## 55 -6.80629190 0.14560400 0.444791172 0.759540892
## 56 -5.52235943 -0.72358314 1.712045515 0.186521461
## 57 -7.03946296 0.23434439 0.509450127 0.596038932
## 58 -6.86627634 0.03719328 1.514328211 0.294311384
## 59 -7.34947829 -0.05957433 0.356208018 0.760881669
## 60 -7.37693108 1.16402026 0.696638459 -0.050557581
## 61 -6.70874378 -0.66445349 2.577755416 0.786050224
## 62 -6.60870085 -0.49073922 1.619800330 1.416365408
## 63 -5.76855627 0.33348017 1.740633836 -1.096362495
## 64 -5.32911937 -1.90229463 1.080048968 1.127357251
## 65 -6.52586487 -0.13212860 0.452738610 -0.422253621
## 66 -5.92829648 0.58983226 1.173668585 -0.051645497
## 67 -5.64035663 0.43981512 0.363999440 -0.080265662
## 68 -5.95543138 0.12642484 1.024480552 -0.272632388
## 69 -7.18772380 0.17555220 0.648509483 0.600517638
## 70 -5.98204436 -0.05892031 1.404884002 -0.117243158
## 71 -6.19698673 0.13612097 0.273177916 -0.620912268
## 72 -6.24267771 0.39008768 0.655579790 0.154535459
## 73 -6.67816125 0.18951888 0.486423300 0.552657888
## 74 -5.90258821 -0.72985809 1.113428765 -0.237851115
## 75 -6.00782295 -0.12544740 0.598775583 -0.922231223
## 76 -6.23610060 0.54710056 0.228273246 0.372182986
## 77 -7.43000943 -0.75686919 0.411388692 0.879701296
## 78 -6.35570641 -0.59554473 1.190746886 0.310234996
## 79 -6.54170316 -0.52780607 0.960782470 0.616558292
## 80 -6.59994795 0.20784909 0.605854957 0.252610851
## 81 -6.29264584 0.81274360 1.044689913 -0.785797950
## 82 -5.16031633 0.13463963 1.038912977 0.526315313
## 83 -5.10751561 -0.01826406 1.835392575 0.325951194
## 84 -5.69818412 0.07129022 2.844589364 0.807207780
## 85 -7.25869074 0.42871902 0.503278693 0.357226724
## 86 -6.74235555 0.13095668 0.678563308 0.122043520
## 87 -6.14130228 0.81993563 1.013338239 0.104778055
## 88 -5.16866977 -0.38744445 0.504503146 -0.245925018
## 89 -6.08538349 2.78331030 1.363977984 -1.523125820
## 90 -8.50886298 0.53283480 0.482828209 0.880000116
## 91 -5.91985544 1.48864094 0.338987526 0.500860827
## 92 -7.06007121 0.75894857 0.596500299 0.398481137
## 93 -6.20598994 1.31247356 0.100542736 -0.187360009
## 94 -5.92586825 0.65690568 1.397989920 0.541963565
## 95 -5.97921767 0.22645940 1.183083071 0.717101005
## 96 -6.41403006 0.75994848 0.383081086 -0.004396699
## 97 -6.96051861 -0.11917730 1.370427634 0.109707771
## 98 -7.00982285 0.44145177 0.299971565 -1.044910815
## 99 -7.08534260 0.55757723 1.791731376 -0.380101354
## 100 -6.14667113 1.69497387 1.013056904 -0.323669784
## 101 -5.92717211 0.58681639 1.326355704 0.746928413
## 102 -6.56203601 0.55664339 2.092424762 -0.334239038
## 103 -6.69416505 0.54421386 1.484104883 0.366231757
## 104 -6.24799166 1.50797759 0.023513902 -0.367437637
## 105 -6.24644325 2.18707105 -0.310186290 -0.711718645
## 106 -5.68675778 0.88989018 1.223783997 -0.283730194
## 107 -6.21200317 0.20340718 1.730508110 0.393579693
## 108 -6.11836918 0.34387084 0.804546016 0.013380459
## 109 -6.26505943 0.02643238 1.344751595 -0.195507223
## 110 -5.89816969 0.69867887 0.993380472 -0.055726153
## 111 -5.21035839 0.05502458 0.549239650 1.222462600
## 112 -7.28606268 1.13851516 0.545122011 0.211364406
## 113 -6.09224227 -0.43982643 1.286017870 -0.985115120
## 114 -7.03997026 0.30724311 0.175166189 -0.081530390
## 115 -5.09195686 0.87531275 0.671575636 1.274196738
## 116 -4.40499219 1.05223273 0.684545843 0.692586375
## 117 -6.02851565 2.39359563 1.597577683 0.935109990
## 118 -6.06947867 1.03119199 2.099015138 0.498409926
## 119 -5.14989064 1.29008249 0.599731516 1.019368713
## 120 -6.30920528 0.30701685 0.087527686 1.363497999
## 121 -4.42561680 1.19234186 0.891317236 1.074324525
## 122 -4.70950811 1.45421708 1.302040192 -0.470985388
## 123 -5.55686261 0.55065933 2.045191021 0.052414814
## 124 4.90531106 0.19331661 -1.341693142 1.640762870
## 125 5.16593914 -0.33691291 0.670941920 -0.390829361
## 126 3.36826645 1.51874847 -0.288340625 1.105806703
## 127 1.87316485 1.50080551 -0.389374368 -0.110739662
## 128 3.69704809 -2.11954222 1.130862073 -0.207284260
## 129 2.03199503 2.07879960 1.181153014 -1.070805644
## 130 1.51529511 1.28884727 -0.805129265 1.340989384
## 131 4.44498818 2.56033838 -0.226230504 0.014609343
## 132 2.99886316 1.16797506 -1.349548031 0.909072510
## 133 3.78720966 0.49259765 -2.691023930 1.059398264
## 134 4.43483180 0.84571192 -0.864741353 0.331627862
## 135 5.94381719 0.79160708 -1.802449516 -1.002549928
## 136 3.25195600 -0.51029053 -1.369445660 0.801807957
## 137 2.92368019 1.53083687 -0.042779686 0.634741377
## 138 5.03638938 1.74225678 -2.091282057 -0.208898638
## 139 5.54818895 0.60142306 -0.829563851 1.805165839
## 140 3.75509347 0.90313079 -2.195044513 1.895110647
## 141 3.95163824 -0.20830403 -0.009960962 0.393528223
## 142 2.58085829 1.31767845 -1.336535952 0.008343446
## 143 2.70871190 1.92065408 0.536803175 0.965499573
## 144 2.42017489 1.17178194 1.436668866 1.629089132
## 145 5.04263916 0.46189937 0.625276818 -0.530577579
## 146 3.94844410 0.26592697 0.820661404 0.355791054
## 147 -3.72324547 -0.54358885 -0.625233533 -0.640412321
## 148 -3.35653392 0.86557455 -0.659180643 -0.873954069
## 149 -3.31708512 0.58375319 0.040484987 -0.199965060
## 150 -4.19349312 0.47432706 -0.809581595 1.808510084
## 151 -4.47140501 3.02167823 -0.608371209 1.756906798
## 152 -4.37649829 0.86993279 -1.157097359 0.091375953
## 153 -3.53084183 2.95926116 -0.370822838 0.044507888
## 154 -2.99859137 1.09919941 -0.735515400 -1.766778474
## 155 -3.70240869 0.05303549 -0.182580056 -0.364447554
## 156 -2.79184625 0.13935047 -1.318194035 0.074368511
## 157 -4.42161319 -0.67137238 -1.691389106 -0.938892960
## 158 -3.09329972 0.49280617 -1.498771890 -1.250435353
## 159 -3.14183530 -0.47716872 -0.350209899 -1.755642068
## 160 -2.94776113 1.86082555 -1.936738005 -2.005872901
## 161 -3.57809127 0.09212330 -1.581393388 -2.317188639
## 162 -3.74953124 0.49425151 -0.111107904 -0.370867004
## 163 -3.70274027 1.86291087 -0.539777811 1.664116866
## 164 -4.07901229 -0.61825614 0.063496551 -0.826850908
## 165 -4.69051373 -0.07663725 -1.730266016 -2.022251734
## 166 -5.08981951 -0.54965095 0.586655596 -0.374799128
## 167 -4.89751612 0.74536799 -0.552606455 -1.282892623
## 168 -2.91014453 0.29482004 0.014559338 1.683803031
## 169 -2.78792635 0.66467857 -2.423681050 -0.093212568
## 170 -2.27362325 1.31661000 0.508664222 -0.576998029
## 171 -0.02925292 -0.99453309 2.088430196 0.083646695
## 172 -1.89840140 -2.07877031 -2.899651672 -0.087652985
## 173 -2.04622417 -2.50060599 0.273229568 0.165350348
## 174 -1.51885777 -1.71066058 -0.477310769 0.324765165
## 175 -1.39470835 -1.93594998 -0.809185404 -1.012129723
## 176 -1.03438684 -2.21275187 0.388501018 1.224650424
## 177 -6.25426092 0.26510233 -0.692733228 -0.438930680
## 178 -5.28131975 2.60280682 0.513933568 0.725434001
## 179 -6.57607519 1.01577908 0.927684690 1.277713511
## 180 -7.51634005 0.81047649 0.300641196 -1.495966949
## 181 -6.63260550 1.63554363 -0.435228422 -0.555221595Class:
The first output here shows how our data are classified based on our LDA model
Posterior:
The posterior probabilities shows the probability of each sample being classified in one group over another. These probabilities measure the strength of each classification. If one is substantially greater than the others, that sample is assigned to that group with a high degree of confidence. Here, many of these probabilities are high so these data are confidently assigned to these groups.
x:
This last part shows the discriminant function axis scores for each sample. These scores can be plotted to show distribution of our samples.
There are a number of ways to visualize these discriminant scores
A simple way to do this is to use the ggord() function in the {ggord} package to plot our model.
Why use this package over others?
1) Super easy to use.
2) A spin-off of ggplot and we all love ggplot so you might like this, too.
library(ggplot2)
library(devtools)
install_github('fawda123/ggord')## Skipping install of 'ggord' from a github remote, the SHA1 (c92d629d) has not changed since last install.
## Use `force = TRUE` to force installation#check out https://github.com/fawda123/ggord/blob/master/R/ggord.R to view original code for this package
library(ggord)At it’s base level, ggord() is pretty good.
Check our this blog post if you want to learn more about what ggord can do.
ggord(camel.lda,camels$Genus..2.)
Easy.
But our data points are little big, and it’s difficult to tell how our variables are influencing our data so we’ll mess around with this plot a little bit.
The ggord default function is a little particular so if you want to use this package to make your own plots, you should check out the raw code for the default ggord() plot to see the easiest way to customize the code.
p<-ggord(camel.lda, camels$Genus..2., txt=3, vec_ext=6, size=1 ) #adjusting text size, vector length, and size of the points
p+ ggtitle("DFA of Davis & McHorse 2013 Camelid data") + #adding a title
theme(plot.title = element_text(lineheight=.8, face="bold"))+ #changing the title text
scale_color_brewer(palette="Dark2") + #changing the palette
theme_minimal() #adding a minimal theme because I like it more 
How does this compare to a PCA plot?
library(FactoMineR) #add the package with PCA() to our library
oo <- PCA(camels[, 8:15], graph = FALSE) #running our PCA model
pca.<-ggord(oo, camels$Genus..2., txt=3, vec_ext=3, size=1 ) #plot our principal components as we did with our linear discriminants
pca.+ ggtitle("PCA of Davis & McHorse Camelid data") + theme(plot.title = element_text(lineheight=.8, face="bold"))+ scale_color_brewer(palette="Dark2") + theme_minimal()
There is more overlap between groups in our PCA plot than in our DFA plot. This is to be expected - recall that PCA tries to retain variability in the data while DFA retains between group variance. Also note that PC1 explains slightly more variance among our data than LD1 does among our groups.
Another nice way to visualize this data is by using the {plotly} package.
Allows us to look at 3 discriminant scores at the same time
Also The plotly package rocks and has very cool interactive visualizations and an online interactive plot making tool
library(plotly)##
## Attaching package: 'plotly'## The following object is masked from 'package:ggplot2':
##
## last_plot## The following object is masked from 'package:MASS':
##
## select## The following object is masked from 'package:stats':
##
## filter## The following object is masked from 'package:graphics':
##
## layoutp<-plot_ly(camels, x=camel.p$x[,1], y=camel.p$x[,2], z=camel.p$x[,3]) %>% #indicate what your axes are
add_markers ( #add markers and a hover label
text=~paste(paste("LD1:",camel.p$x[,1]), paste("LD2:",camel.p$x[,2]), paste("LD3:",camel.p$x[,3]) , paste("Genus:", camels$Genus..2.), sep="<br />"), #what each element in your hover label will be and how you will separate them (if sep= "<br />" wasn't here they would all print on one long hover label)
color=~camels$Genus..2., colors="Set1",symbol=I("circle"), size=I(6), hoverinfo="text") %>% #how you will define points (by genus), what colors you will use, the symbol and symbol size of each point, and what what info will be shown in the hover label (in this case it is what we just "text" as)
layout(scene=list(xaxis=list(title="LD1"),
yaxis=list(title="LD2"),
zaxis=list(title="LD3"))) #axes labels
pWe have it plotted, what now?
We are now interested in how accurately our discriminant function analysis classified our samples into our groups.
To assess this, we need to compare our predicted groups with our user-specified groups.
cam <- table(camels$Genus..2., camel.p$class) #make a table of our original group assignments and predicted group assignments
cam #here the rows are original group assignments and the columns are predicted group assignments##
## Aepycamelus Alforjas Hemiauchenia Megatylopus Procamelus
## Aepycamelus 44 0 0 4 0
## Alforjas 0 23 1 0 0
## Hemiauchenia 0 0 80 0 0
## Megatylopus 1 0 0 22 0
## Procamelus 0 0 0 0 6From here we can quantify the overall predictive accuracy of our LDA model.
sum(cam[row(cam) == col(cam)]) / sum(cam) #summing the number of samples that were classified in the prediction as in our original data divided by the total number of samples## [1] 0.9668508Since we’ll be doing this again, it will be useful to have a function that generates a confusion matrix and calculates overall accuracy. I didn’t write the following function. I got it from here and only modified it slightly but I like the output it generates. It sure makes it easy to spit out information about the accuracy of your model (as we did in multiple lines of code above but now we can do it in one line).
#this is a function that gives us the overally accuracy of our model, our prior probabilities, and a table demonstrating what proportion of our groups were classified in their correct/incorrect groups.
library(lattice)
confusion <- function(actual, predicted, names = NULL, printit = TRUE, prior = NULL) { #names and priors are null unless otherwise specified; R should print our output
if (is.null(names))
names <- levels(actual)
tab <- table(actual, predicted)
acctab <- t(apply(tab, 1, function(x) x/sum(x)))
dimnames(acctab) <- list(Actual = names, "Predicted" = names)
if (is.null(prior)) {
relnum <- table(actual)
prior <- relnum/sum(relnum)
acc <- sum(tab[row(tab) == col(tab)])/sum(tab)
}
else {
acc <- sum(prior * diag(acctab))
names(prior) <- names
}
if (printit)
print(round(c("Overall accuracy" = acc, "Prior frequency" = prior),
4))
if (printit) {
cat("\nConfusion matrix", "\n")
print(round(acctab, 4))
}
invisible(acctab)
}
original <- confusion(camels$Genus..2., camel.p$class,prior= c(1,1,1,1,1)/5) #generating our confusion matrix for our camel lda model## Overall accuracy Prior frequency.Aepycamelus
## 0.9663 0.2000
## Prior frequency.Alforjas Prior frequency.Hemiauchenia
## 0.2000 0.2000
## Prior frequency.Megatylopus Prior frequency.Procamelus
## 0.2000 0.2000
##
## Confusion matrix
## Predicted
## Actual Aepycamelus Alforjas Hemiauchenia Megatylopus Procamelus
## Aepycamelus 0.9167 0.0000 0.0000 0.0833 0
## Alforjas 0.0000 0.9583 0.0417 0.0000 0
## Hemiauchenia 0.0000 0.0000 1.0000 0.0000 0
## Megatylopus 0.0435 0.0000 0.0000 0.9565 0
## Procamelus 0.0000 0.0000 0.0000 0.0000 1So what we just did used the resubstitution error method - so just a reminder - this is how well the samples are classified when all the samples are used to generate our models.
But what if we want to use jackknifing?
Reminder: Jackknifed validation (or leave-out-one cross-validation) excludes one of our samples, generates a discriminant function with the remaining samples, uses this discriminant function to reclassify our excluded sample, and repeats this for each sample in our data set.
This can be done simply by indicating CV=TRUE in our lda() call.
knifed_camel <- lda(Genus..2. ~ LM + TD + TI + TP + LL + WD + WI + LI, data=camels, prior= c(1,1,1,1,1)/5, na.action="na.omit", CV=TRUE) #CV=TRUE means we're going to jackknife the cuss out of this thing
knifed_camel # show results## $class
## [1] Aepycamelus Aepycamelus Aepycamelus Aepycamelus Aepycamelus
## [6] Aepycamelus Aepycamelus Aepycamelus Aepycamelus Aepycamelus
## [11] Aepycamelus Aepycamelus Aepycamelus Megatylopus Aepycamelus
## [16] Aepycamelus Aepycamelus Megatylopus Aepycamelus Aepycamelus
## [21] Aepycamelus Aepycamelus Aepycamelus Aepycamelus Aepycamelus
## [26] Aepycamelus Aepycamelus Aepycamelus Aepycamelus Megatylopus
## [31] Aepycamelus Aepycamelus Aepycamelus Aepycamelus Aepycamelus
## [36] Aepycamelus Aepycamelus Aepycamelus Aepycamelus Aepycamelus
## [41] Aepycamelus Aepycamelus Aepycamelus Aepycamelus Aepycamelus
## [46] Megatylopus Aepycamelus Aepycamelus Hemiauchenia Hemiauchenia
## [51] Hemiauchenia Hemiauchenia Hemiauchenia Hemiauchenia Hemiauchenia
## [56] Hemiauchenia Hemiauchenia Hemiauchenia Hemiauchenia Hemiauchenia
## [61] Hemiauchenia Hemiauchenia Hemiauchenia Hemiauchenia Hemiauchenia
## [66] Hemiauchenia Hemiauchenia Hemiauchenia Hemiauchenia Hemiauchenia
## [71] Hemiauchenia Hemiauchenia Hemiauchenia Hemiauchenia Hemiauchenia
## [76] Hemiauchenia Hemiauchenia Hemiauchenia Hemiauchenia Hemiauchenia
## [81] Hemiauchenia Hemiauchenia Hemiauchenia Hemiauchenia Hemiauchenia
## [86] Hemiauchenia Hemiauchenia Hemiauchenia Hemiauchenia Hemiauchenia
## [91] Hemiauchenia Hemiauchenia Hemiauchenia Hemiauchenia Hemiauchenia
## [96] Hemiauchenia Hemiauchenia Hemiauchenia Hemiauchenia Hemiauchenia
## [101] Hemiauchenia Hemiauchenia Hemiauchenia Hemiauchenia Hemiauchenia
## [106] Hemiauchenia Hemiauchenia Hemiauchenia Hemiauchenia Hemiauchenia
## [111] Hemiauchenia Hemiauchenia Hemiauchenia Hemiauchenia Hemiauchenia
## [116] Alforjas Hemiauchenia Hemiauchenia Hemiauchenia Hemiauchenia
## [121] Hemiauchenia Hemiauchenia Hemiauchenia Megatylopus Aepycamelus
## [126] Megatylopus Megatylopus Megatylopus Megatylopus Megatylopus
## [131] Megatylopus Megatylopus Megatylopus Megatylopus Aepycamelus
## [136] Megatylopus Megatylopus Megatylopus Megatylopus Megatylopus
## [141] Megatylopus Megatylopus Megatylopus Megatylopus Megatylopus
## [146] Megatylopus Alforjas Alforjas Alforjas Alforjas
## [151] Alforjas Alforjas Alforjas Alforjas Alforjas
## [156] Alforjas Alforjas Alforjas Alforjas Alforjas
## [161] Alforjas Alforjas Alforjas Alforjas Alforjas
## [166] Hemiauchenia Alforjas Alforjas Alforjas Alforjas
## [171] Procamelus Procamelus Procamelus Procamelus Procamelus
## [176] Procamelus Hemiauchenia Hemiauchenia Hemiauchenia Hemiauchenia
## [181] Hemiauchenia
## Levels: Aepycamelus Alforjas Hemiauchenia Megatylopus Procamelus
##
## $posterior
## Aepycamelus Alforjas Hemiauchenia Megatylopus Procamelus
## 1 9.999649e-01 6.559234e-30 2.044637e-44 3.510405e-05 1.533861e-18
## 2 9.999865e-01 1.601968e-31 9.423372e-46 1.354178e-05 6.903964e-20
## 3 1.000000e+00 3.111942e-42 9.947166e-59 7.555982e-10 4.875788e-28
## 4 9.999998e-01 2.351039e-39 6.745386e-55 1.811409e-07 1.417319e-28
## 5 9.999825e-01 1.638383e-30 3.921160e-46 1.745387e-05 9.073691e-22
## 6 9.952176e-01 6.255639e-28 1.080748e-40 4.782434e-03 6.111753e-18
## 7 9.985970e-01 2.641379e-22 3.497555e-35 1.403047e-03 1.089304e-12
## 8 9.999996e-01 5.718942e-38 2.517187e-54 3.529948e-07 4.308590e-27
## 9 9.999252e-01 5.048689e-31 8.054680e-47 7.478916e-05 2.504907e-23
## 10 8.605713e-01 2.662432e-19 9.393703e-34 1.394287e-01 7.753889e-13
## 11 9.854172e-01 7.450862e-26 1.528825e-39 1.458280e-02 9.800921e-17
## 12 6.279045e-01 9.710716e-23 1.390348e-33 3.720955e-01 1.922291e-14
## 13 8.786958e-01 6.069400e-22 9.629517e-35 1.213042e-01 2.885461e-12
## 14 1.055598e-02 2.531958e-16 8.329366e-27 9.894440e-01 2.753146e-08
## 15 1.000000e+00 2.199891e-39 7.251675e-54 7.719679e-09 5.387779e-28
## 16 9.999981e-01 3.259343e-32 1.579617e-45 1.944653e-06 3.998732e-19
## 17 9.463178e-01 7.426431e-23 7.025875e-37 5.368218e-02 9.321648e-17
## 18 2.716876e-01 1.319141e-18 3.991644e-31 7.283124e-01 3.879389e-12
## 19 9.998340e-01 4.525974e-27 3.076968e-41 1.660392e-04 1.297060e-17
## 20 9.999797e-01 3.631591e-32 7.752128e-48 2.029205e-05 2.406217e-26
## 21 9.987303e-01 2.474663e-28 6.093592e-43 1.269662e-03 1.475981e-20
## 22 9.993455e-01 1.079569e-27 1.399204e-39 6.545030e-04 4.437397e-18
## 23 9.999794e-01 3.111583e-27 4.860840e-41 2.058051e-05 3.298095e-18
## 24 9.800802e-01 3.659507e-21 5.114399e-33 1.991980e-02 1.369543e-12
## 25 9.736372e-01 5.901171e-25 8.788819e-39 2.636280e-02 9.195154e-17
## 26 9.999954e-01 5.049984e-32 2.045431e-45 4.594977e-06 1.944672e-23
## 27 9.996607e-01 4.548074e-27 1.615950e-39 3.393081e-04 3.147849e-17
## 28 9.963244e-01 1.197172e-25 8.018213e-40 3.675579e-03 1.787726e-14
## 29 9.999984e-01 7.350628e-34 4.082050e-49 1.560735e-06 1.889307e-21
## 30 1.962757e-04 2.624846e-12 6.756100e-22 9.998036e-01 8.236873e-08
## 31 9.997305e-01 1.383073e-28 2.594164e-42 2.695110e-04 4.432794e-18
## 32 9.999928e-01 2.270784e-28 2.787629e-41 7.167161e-06 3.324845e-15
## 33 9.999953e-01 7.851126e-30 3.060721e-42 4.700104e-06 4.077861e-18
## 34 9.999950e-01 4.669367e-35 9.494146e-50 4.989828e-06 1.202920e-23
## 35 9.999930e-01 2.582847e-34 3.175251e-50 7.019976e-06 1.379529e-23
## 36 9.857106e-01 1.258967e-28 7.307088e-44 1.428942e-02 1.778809e-19
## 37 9.999997e-01 1.377741e-40 1.783409e-58 3.409720e-07 4.309703e-28
## 38 9.995190e-01 9.441701e-27 6.643192e-40 4.809572e-04 4.610076e-17
## 39 9.999327e-01 2.069951e-27 6.422381e-39 6.734888e-05 9.359501e-18
## 40 9.997913e-01 7.700770e-28 2.055194e-40 2.087124e-04 1.956131e-18
## 41 9.377098e-01 6.951873e-24 3.263465e-37 6.229019e-02 8.273463e-16
## 42 9.999959e-01 5.145546e-35 7.090605e-49 4.067528e-06 5.442337e-23
## 43 9.998949e-01 5.222066e-30 1.641257e-45 1.051481e-04 2.196114e-20
## 44 9.999774e-01 1.902796e-30 4.419875e-45 2.262191e-05 9.482793e-18
## 45 9.997895e-01 2.194354e-26 3.507219e-38 2.104958e-04 7.406871e-17
## 46 1.250130e-03 3.094000e-12 1.218644e-19 9.987497e-01 1.321781e-07
## 47 9.941204e-01 2.130953e-22 6.274428e-35 5.879611e-03 3.595492e-11
## 48 1.000000e+00 4.481036e-41 3.427577e-54 2.309125e-09 8.927695e-30
## 49 6.142989e-41 1.308215e-02 9.869177e-01 4.273496e-22 1.137098e-07
## 50 1.297522e-42 6.577986e-03 9.934219e-01 1.160556e-22 1.051949e-07
## 51 1.578705e-41 1.352441e-02 9.864755e-01 3.012255e-22 4.088187e-08
## 52 2.780687e-46 2.817609e-03 9.971824e-01 8.130208e-26 1.190240e-08
## 53 1.965210e-36 3.697521e-01 6.300435e-01 4.351182e-19 2.044398e-04
## 54 2.997042e-42 1.018691e-02 9.898129e-01 3.249810e-23 1.522030e-07
## 55 4.181927e-45 2.672711e-03 9.973272e-01 9.518124e-25 4.213160e-08
## 56 3.526014e-37 1.214699e-02 9.878026e-01 4.426991e-20 5.044302e-05
## 57 1.338668e-46 1.454637e-03 9.985454e-01 7.274738e-26 9.730249e-09
## 58 1.699850e-45 5.035376e-04 9.994965e-01 9.077728e-26 1.202987e-08
## 59 1.544376e-48 7.195324e-04 9.992805e-01 3.337898e-27 4.984014e-09
## 60 7.005641e-49 7.327739e-04 9.992672e-01 1.793520e-27 1.418195e-10
## 61 1.186075e-44 8.129007e-05 9.999187e-01 8.050018e-26 4.364851e-08
## 62 5.141878e-44 3.937391e-04 9.996061e-01 1.385220e-24 1.365237e-07
## 63 1.242688e-38 1.477581e-02 9.852230e-01 3.238750e-21 1.141311e-06
## 64 1.157551e-35 3.116081e-02 9.625100e-01 9.513067e-19 6.329225e-03
## 65 4.866626e-43 1.033798e-02 9.896617e-01 9.974148e-24 3.327662e-07
## 66 8.396337e-40 1.330429e-02 9.866952e-01 2.118654e-21 4.909151e-07
## 67 4.769853e-38 9.282854e-02 9.071651e-01 8.693389e-20 6.340976e-06
## 68 8.179224e-40 1.652169e-02 9.834768e-01 1.518516e-21 1.496661e-06
## 69 1.609531e-47 7.720244e-04 9.992280e-01 1.297708e-26 4.568309e-09
## 70 6.402485e-40 8.025111e-03 9.919734e-01 7.869148e-22 1.538001e-06
## 71 4.720483e-41 3.787178e-02 9.621271e-01 3.564320e-22 1.094171e-06
## 72 1.353586e-41 1.177026e-02 9.882294e-01 1.901059e-22 2.988081e-07
## 73 2.896089e-44 3.943328e-03 9.960566e-01 3.198868e-24 6.858362e-08
## 74 2.663125e-39 1.474785e-02 9.852382e-01 1.794507e-21 1.397859e-05
## 75 7.645206e-40 4.212704e-02 9.578693e-01 1.247274e-21 3.663951e-06
## 76 1.389826e-41 2.122334e-02 9.787763e-01 4.098904e-22 3.360777e-07
## 77 5.364504e-49 4.444428e-04 9.995555e-01 9.412220e-28 1.725825e-08
## 78 3.832643e-42 2.989592e-03 9.970094e-01 2.209071e-23 1.032296e-06
## 79 2.502760e-43 2.252107e-03 9.977474e-01 5.245558e-24 4.454156e-07
## 80 9.896051e-44 4.711143e-03 9.952888e-01 5.469209e-24 8.453924e-08
## 81 7.606682e-42 1.040415e-02 9.895958e-01 5.702388e-23 5.665090e-08
## 82 1.846017e-35 7.516488e-02 9.247712e-01 4.066921e-18 6.394040e-05
## 83 4.333890e-35 2.859147e-02 9.713515e-01 2.552754e-18 5.699557e-05
## 84 1.315461e-38 8.849666e-04 9.991141e-01 2.807711e-21 8.893939e-07
## 85 5.626189e-48 9.757016e-04 9.990243e-01 7.517710e-27 2.028603e-09
## 86 1.450388e-44 3.073345e-03 9.969266e-01 1.087836e-24 4.627869e-08
## 87 4.060630e-41 9.468622e-03 9.905313e-01 3.702630e-22 1.187464e-07
## 88 3.292320e-35 2.218991e-01 7.777610e-01 4.536646e-18 3.398893e-04
## 89 7.586970e-41 2.580686e-02 9.741931e-01 5.535139e-22 7.027699e-10
## 90 5.477194e-57 2.474456e-05 9.999753e-01 6.632280e-33 1.937257e-12
## 91 5.920235e-40 4.258510e-02 9.574148e-01 1.173927e-20 1.463094e-07
## 92 8.826367e-47 1.435496e-03 9.985645e-01 6.224523e-26 2.221231e-09
## 93 1.873821e-41 4.303119e-02 9.569687e-01 7.010762e-22 7.179417e-08
## 94 6.000030e-40 6.649987e-03 9.933497e-01 2.017600e-21 3.246374e-07
## 95 3.509402e-40 6.936465e-03 9.930627e-01 1.437038e-21 8.720628e-07
## 96 1.209869e-42 1.346792e-02 9.865320e-01 5.348575e-23 7.173154e-08
## 97 5.032157e-46 5.427683e-04 9.994572e-01 3.490442e-26 1.298502e-08
## 98 4.483221e-46 6.083933e-03 9.939161e-01 7.916145e-26 9.045463e-09
## 99 6.393982e-47 2.860753e-04 9.997139e-01 5.635008e-27 8.059173e-10
## 100 2.884800e-41 1.348084e-02 9.865191e-01 4.075664e-22 1.386424e-08
## 101 5.797791e-40 6.581275e-03 9.934183e-01 2.336443e-21 4.150650e-07
## 102 1.129348e-43 6.715616e-04 9.993284e-01 8.822364e-25 8.101000e-09
## 103 1.519797e-44 8.560932e-04 9.991439e-01 7.026563e-25 8.534923e-09
## 104 1.072824e-41 5.228114e-02 9.477188e-01 5.364178e-22 3.979909e-08
## 105 9.474271e-42 1.204094e-01 8.795906e-01 9.528723e-22 1.027816e-08
## 106 2.099123e-38 2.756687e-02 9.724324e-01 2.237055e-20 7.554422e-07
## 107 1.500976e-41 1.894491e-03 9.981053e-01 6.075454e-23 1.646589e-07
## 108 7.582965e-41 1.350612e-02 9.864934e-01 4.886870e-22 5.112412e-07
## 109 1.159052e-41 4.273718e-03 9.957260e-01 4.612340e-23 3.014933e-07
## 110 1.236022e-39 1.937390e-02 9.806256e-01 3.710685e-21 5.307770e-07
## 111 9.363143e-36 9.675419e-02 9.031412e-01 6.230633e-18 1.046118e-04
## 112 2.521766e-48 1.019972e-03 9.989800e-01 6.347261e-27 2.842513e-10
## 113 2.366821e-40 1.106565e-02 9.889320e-01 1.833082e-22 2.380410e-06
## 114 2.139326e-46 3.787905e-03 9.962121e-01 9.783521e-26 1.222738e-08
## 115 3.000408e-35 1.185013e-01 8.814743e-01 2.358459e-17 2.437994e-05
## 116 1.029384e-31 5.310740e-01 4.686814e-01 6.615406e-15 2.445863e-04
## 117 5.020746e-41 3.798039e-03 9.962020e-01 1.254353e-21 2.158847e-09
## 118 6.276554e-41 1.591499e-03 9.984085e-01 2.212176e-22 2.948090e-08
## 119 1.436218e-35 1.420163e-01 8.579762e-01 1.685344e-17 7.559336e-06
## 120 3.911344e-42 1.287187e-02 9.871276e-01 3.385378e-22 5.154126e-07
## 121 1.233831e-31 4.156203e-01 5.841918e-01 9.209767e-15 1.878837e-04
## 122 4.068919e-33 2.809416e-01 7.190414e-01 2.107899e-16 1.695986e-05
## 123 9.430770e-38 7.999302e-03 9.919994e-01 2.645948e-20 1.284451e-06
## 124 1.103292e-02 5.879186e-17 1.192947e-28 9.889671e-01 6.307656e-10
## 125 7.166374e-01 1.052470e-17 1.134781e-28 2.833626e-01 8.496365e-10
## 126 3.165664e-05 9.877481e-12 5.947841e-21 9.999683e-01 7.329243e-08
## 127 1.991745e-07 2.927367e-06 2.817181e-14 9.996289e-01 3.679478e-04
## 128 6.533605e-02 5.842675e-12 1.311171e-20 9.332813e-01 1.382682e-03
## 129 3.469312e-06 1.888394e-06 8.393290e-14 9.999074e-01 8.722890e-05
## 130 8.970195e-09 9.679287e-06 2.452483e-13 9.981685e-01 1.821859e-03
## 131 2.275639e-03 5.457915e-15 8.692626e-26 9.977244e-01 1.748185e-11
## 132 3.270257e-06 1.950705e-10 4.857214e-20 9.999959e-01 8.606808e-07
## 133 3.176078e-05 1.040701e-12 3.836468e-24 9.999682e-01 8.824667e-08
## 134 3.298825e-03 7.199594e-15 6.320985e-26 9.967012e-01 1.984768e-09
## 135 9.518914e-01 2.632940e-21 2.847268e-36 4.810856e-02 2.472799e-14
## 136 4.953016e-05 6.464711e-11 9.715013e-21 9.999172e-01 3.328766e-05
## 137 9.964674e-06 3.523602e-10 7.007524e-19 9.999892e-01 8.236874e-07
## 138 8.661018e-03 1.190726e-16 9.876948e-30 9.913390e-01 5.021810e-12
## 139 1.220144e-01 1.628443e-19 1.188388e-31 8.779856e-01 4.956328e-12
## 140 1.712322e-05 5.046468e-13 7.357370e-24 9.999828e-01 2.801336e-08
## 141 5.561998e-03 3.963657e-13 6.883105e-23 9.944370e-01 9.740002e-07
## 142 1.187606e-06 8.391131e-09 3.476934e-18 9.999924e-01 6.441958e-06
## 143 4.530429e-06 1.241473e-09 1.254528e-17 9.999945e-01 9.935251e-07
## 144 5.407505e-06 1.202947e-08 2.195250e-15 9.999246e-01 7.001282e-05
## 145 4.378608e-01 5.659767e-17 6.626623e-28 5.621392e-01 3.153738e-10
## 146 5.662409e-03 2.288701e-13 1.466393e-22 9.943374e-01 2.095539e-07
## 147 6.646770e-28 9.381032e-01 1.623816e-02 6.034434e-13 4.565868e-02
## 148 1.173835e-26 9.945164e-01 3.479916e-03 8.240855e-12 2.003665e-03
## 149 7.663305e-26 9.706679e-01 1.809782e-02 2.895683e-11 1.123424e-02
## 150 1.238079e-30 8.641752e-01 1.325122e-01 1.838941e-13 3.312564e-03
## 151 5.767366e-33 6.369422e-01 3.630562e-01 3.563824e-14 1.614370e-06
## 152 5.805751e-32 9.601655e-01 3.959420e-02 8.816802e-15 2.402649e-04
## 153 9.003531e-28 9.855130e-01 1.447462e-02 1.218402e-11 1.241010e-05
## 154 4.549909e-25 9.976879e-01 7.171961e-04 5.546007e-11 1.594855e-03
## 155 8.140078e-28 9.625103e-01 2.640654e-02 9.354542e-13 1.108313e-02
## 156 3.997553e-24 9.441425e-01 5.008936e-04 8.358528e-10 5.535660e-02
## 157 4.181275e-32 9.807197e-01 1.386481e-02 1.449299e-15 5.415493e-03
## 158 1.392966e-25 9.925921e-01 4.324846e-04 4.466156e-11 6.975408e-03
## 159 5.019876e-25 9.143819e-01 2.522040e-03 1.671460e-11 8.309610e-02
## 160 2.376784e-25 9.997612e-01 7.702843e-05 1.192792e-10 1.617742e-04
## 161 5.323889e-28 9.961952e-01 7.510109e-04 3.396016e-13 3.053743e-03
## 162 3.875753e-28 9.659364e-01 3.074935e-02 7.218324e-13 3.314277e-03
## 163 2.454639e-28 9.480166e-01 5.163288e-02 1.145527e-11 3.505326e-04
## 164 2.594372e-29 8.550092e-01 1.156595e-01 3.511476e-14 2.933129e-02
## 165 7.641320e-34 9.847876e-01 1.485525e-02 6.358823e-17 3.571412e-04
## 166 3.704205e-35 1.365943e-01 8.628088e-01 3.094405e-18 5.968832e-04
## 167 1.717562e-34 8.096734e-01 1.902732e-01 3.575023e-17 5.342446e-05
## 168 9.282051e-24 8.225437e-01 1.702333e-02 2.902250e-09 1.604329e-01
## 169 1.201986e-24 9.891443e-01 7.447179e-05 1.002498e-09 1.078125e-02
## 170 7.511923e-21 9.754572e-01 1.870440e-03 6.162511e-08 2.267232e-02
## 171 1.137026e-09 1.722857e-02 2.237823e-06 4.974914e-02 9.330201e-01
## 172 3.967003e-21 2.031403e-01 8.326963e-07 2.978880e-08 7.968589e-01
## 173 3.451476e-21 6.894674e-03 1.118918e-05 9.228102e-10 9.930941e-01
## 174 1.085982e-18 1.924787e-02 2.196291e-06 1.826509e-07 9.807498e-01
## 175 1.468064e-18 9.428294e-03 2.116784e-07 5.769376e-08 9.905714e-01
## 176 2.686315e-17 3.708632e-04 9.186237e-08 3.602024e-07 9.996287e-01
## 177 2.286623e-41 1.355152e-01 8.644831e-01 7.959830e-22 1.681372e-06
## 178 1.931042e-36 1.751052e-01 8.248946e-01 9.192416e-18 1.790896e-07
## 179 4.617394e-44 1.834927e-03 9.981651e-01 8.407333e-24 8.859799e-09
## 180 1.299002e-49 2.287236e-03 9.977128e-01 2.354956e-28 2.339305e-10
## 181 4.567787e-44 5.530171e-02 9.446983e-01 2.017839e-23 6.715686e-09
##
## $terms
## Genus..2. ~ LM + TD + TI + TP + LL + WD + WI + LI
## attr(,"variables")
## list(Genus..2., LM, TD, TI, TP, LL, WD, WI, LI)
## attr(,"factors")
## LM TD TI TP LL WD WI LI
## Genus..2. 0 0 0 0 0 0 0 0
## LM 1 0 0 0 0 0 0 0
## TD 0 1 0 0 0 0 0 0
## TI 0 0 1 0 0 0 0 0
## TP 0 0 0 1 0 0 0 0
## LL 0 0 0 0 1 0 0 0
## WD 0 0 0 0 0 1 0 0
## WI 0 0 0 0 0 0 1 0
## LI 0 0 0 0 0 0 0 1
## attr(,"term.labels")
## [1] "LM" "TD" "TI" "TP" "LL" "WD" "WI" "LI"
## attr(,"order")
## [1] 1 1 1 1 1 1 1 1
## attr(,"intercept")
## [1] 1
## attr(,"response")
## [1] 1
## attr(,".Environment")
## <environment: R_GlobalEnv>
## attr(,"predvars")
## list(Genus..2., LM, TD, TI, TP, LL, WD, WI, LI)
## attr(,"dataClasses")
## Genus..2. LM TD TI TP LL WD
## "factor" "numeric" "numeric" "numeric" "numeric" "numeric" "numeric"
## WI LI
## "numeric" "numeric"
##
## $call
## lda(formula = Genus..2. ~ LM + TD + TI + TP + LL + WD + WI +
## LI, data = camels, prior = c(1, 1, 1, 1, 1)/5, CV = TRUE,
## na.action = "na.omit")
##
## $xlevels
## named list()Class:
Classification of each sample using our jackknifing method. Note: since the jackknifing method includes reclassifying our sample based on the model generated when that sample was removed, we don’t use the predict() function.
Posterior:
Posterior probabilities of each sample being categorized within each group. Again, the strength of the new classification is indicated by these probabilities.
Terms:
Basically a summary of the varaiables in your model.
Call:
The formula that was fitted.
One final step:
Compare the accuracy of our jackknifed model with our non-jackknifed model.
# Assess the accuracy of the prediction
confusion(camels$Genus..2., knifed_camel$class,prior= c(1,1,1,1,1)/5)## Overall accuracy Prior frequency.Aepycamelus
## 0.9551 0.2000
## Prior frequency.Alforjas Prior frequency.Hemiauchenia
## 0.2000 0.2000
## Prior frequency.Megatylopus Prior frequency.Procamelus
## 0.2000 0.2000
##
## Confusion matrix
## Predicted
## Actual Aepycamelus Alforjas Hemiauchenia Megatylopus Procamelus
## Aepycamelus 0.9167 0.0000 0.0000 0.0833 0
## Alforjas 0.0000 0.9583 0.0417 0.0000 0
## Hemiauchenia 0.0000 0.0125 0.9875 0.0000 0
## Megatylopus 0.0870 0.0000 0.0000 0.9130 0
## Procamelus 0.0000 0.0000 0.0000 0.0000 1original #look at our original model again## Predicted
## Actual Aepycamelus Alforjas Hemiauchenia Megatylopus Procamelus
## Aepycamelus 0.91666667 0.0000000 0.00000000 0.08333333 0
## Alforjas 0.00000000 0.9583333 0.04166667 0.00000000 0
## Hemiauchenia 0.00000000 0.0000000 1.00000000 0.00000000 0
## Megatylopus 0.04347826 0.0000000 0.00000000 0.95652174 0
## Procamelus 0.00000000 0.0000000 0.00000000 0.00000000 1Generally speaking, we could expect our jackknifed model to not be nearly as accurate as our non-jackknifed model.
Iris Dataset
Fisher’s 1936 Paper
Ronald Fisher thinking about discriminant analysis…
Now with the iris data set we can repeat what we did with the camelids and run a linear discriminant analysis on our data.
ldafit <- lda(Species ~ ., iris, prior = rep(1, 3)/3)
ldafit ## Call:
## lda(Species ~ ., data = iris, prior = rep(1, 3)/3)
##
## Prior probabilities of groups:
## setosa versicolor virginica
## 0.3333333 0.3333333 0.3333333
##
## Group means:
## Sepal.Length Sepal.Width Petal.Length Petal.Width
## setosa 5.006 3.428 1.462 0.246
## versicolor 5.936 2.770 4.260 1.326
## virginica 6.588 2.974 5.552 2.026
##
## Coefficients of linear discriminants:
## LD1 LD2
## Sepal.Length 0.8293776 0.02410215
## Sepal.Width 1.5344731 2.16452123
## Petal.Length -2.2012117 -0.93192121
## Petal.Width -2.8104603 2.83918785
##
## Proportion of trace:
## LD1 LD2
## 0.9912 0.0088irispredict<-predict(ldafit)
confusion(iris$Species, irispredict$class)## Overall accuracy Prior frequency.setosa
## 0.9800 0.3333
## Prior frequency.versicolor Prior frequency.virginica
## 0.3333 0.3333
##
## Confusion matrix
## Predicted
## Actual setosa versicolor virginica
## setosa 1 0.00 0.00
## versicolor 0 0.96 0.04
## virginica 0 0.02 0.98ggord(ldafit, iris$Species)
<- Simple!
Quadratic Discriminant Analysis with 3 groups
Reminder: Quadratic Discriminant Analysis is used when we can’t assume homogeneity of our variance-covariance matrices. Again we’ll use the {Mass} package but this time use the qda() function.
qdafit <- qda(Species ~ Sepal.Length+Sepal.Width+Petal.Length+Petal.Width, iris, prior = rep(1,3)/3)
qdafit## Call:
## qda(Species ~ Sepal.Length + Sepal.Width + Petal.Length + Petal.Width,
## data = iris, prior = rep(1, 3)/3)
##
## Prior probabilities of groups:
## setosa versicolor virginica
## 0.3333333 0.3333333 0.3333333
##
## Group means:
## Sepal.Length Sepal.Width Petal.Length Petal.Width
## setosa 5.006 3.428 1.462 0.246
## versicolor 5.936 2.770 4.260 1.326
## virginica 6.588 2.974 5.552 2.026Once again we get the prior probabilities of groups and group means. Now we’ll use our confusion() function to get overall accuracy of our QDA model:
confusion(iris$Species, predict(qdafit)$class)## Overall accuracy Prior frequency.setosa
## 0.9800 0.3333
## Prior frequency.versicolor Prior frequency.virginica
## 0.3333 0.3333
##
## Confusion matrix
## Predicted
## Actual setosa versicolor virginica
## setosa 1 0.00 0.00
## versicolor 0 0.96 0.04
## virginica 0 0.02 0.98Some cool exploratory visualization tools
The {klaR} package allows us to look at how successful our classification methods are for every combination of 2 variables.
#Exploratory graph for LDA or QDA
library(klaR)
partimat(Species ~ Sepal.Length+Sepal.Width+Petal.Length+Petal.Width,data=iris,method="lda")
Similarly we can use the pairs() function in the basic graphics package to look at combinations of our variables.
pairs(iris[c("Sepal.Length","Sepal.Width","Petal.Length", "Petal.Width")], main="Scatterplot matrix for iris data", pch=22,
bg=c("red", "yellow", "blue")[unclass(iris$Species)])
Again, if we want to look at 3 axes at the same time 3D plots are useful.
The {scatterplot3d} package has nice visualization of scatterplots.
While they are not interactive and therefore fabulous, they are probably better for print, i.e. boring.
Here I’ll demonstrate this with the Sepal/Petal Widths/Lengths although you could also do one of these plots to show your LD scores as we did above with the camelids.
library(scatterplot3d)
#plot the following 4 plots on the same page
par(mfrow = c(2, 2)) #number of rows
mar0 = c(2, 3, 2, 3) #dimensions for each plot
scatterplot3d(iris[, 1], iris[, 2], iris[, 3], mar = mar0, color = c("red", #specify which columns of the data set you want to plot, what the margins should be, and what color
"yellow", "blue")[iris$Species], pch = 19) #what you should separate colors by (species) and the size of the points
scatterplot3d(iris[, 2], iris[, 3], iris[, 4], mar = mar0, color = c("red",
"yellow", "blue")[iris$Species], pch = 19)
scatterplot3d(iris[, 3], iris[, 4], iris[, 1], mar = mar0, color = c("red",
"yellow", "blue")[iris$Species], pch = 19)
scatterplot3d(iris[, 4], iris[, 1], iris[, 2], mar = mar0, color = c("red",
"yellow", "blue")[iris$Species], pch = 19)
Finally, because javascript is terrific, here’s a {plotly} plot of our LD scores for the iris data set:
p <- plot_ly(data = iris, x = ~irispredict$x[,1], y = ~irispredict$x[,2], color = ~Species,
text=~paste("Species:",Species,"LDA1:", irispredict$x[,1], "LDA2:",irispredict$x[,2], sep="<br />"),
hoverinfo="text")
p## No trace type specified:
## Based on info supplied, a 'scatter' trace seems appropriate.
## Read more about this trace type -> https://plot.ly/r/reference/#scatter## No scatter mode specifed:
## Setting the mode to markers
## Read more about this attribute -> https://plot.ly/r/reference/#scatter-modeOn these 2D plots you can use your mouse to zoom in on certain parts of your plot and double click to zoom back out. COOL!
